Solve the problem.
-Suppose that two microphones 2,800 m apart at points A = (1,400, 0) and B = (-1,400, 0) detect the sound of a rifle shot. The time difference between the sound detected at A and the sound detected at B is 2 sec. If sound travels at approximately 330 m/sec, find an equation of the hyperbola with foci
At A and B defining the points where the shooter may be located.
A) $\frac { x ^ { 2 } } { 330 ^ { 2 } } - \frac { y ^ { 2 } } { 1,361 ^ { 2 } } = 1$
B) $\frac { x ^ { 2 } } { 1,400 ^ { 2 } } - \frac { y ^ { 2 } } { 330 ^ { 2 } } = 1$
C) $\frac { x ^ { 2 } } { 2,800 ^ { 2 } } - \frac { y ^ { 2 } } { 2,721 ^ { 2 } } = 1$
D) $\frac { y ^ { 2 } } { 1,361 ^ { 2 } } - \frac { x ^ { 2 } } { 330 ^ { 2 } } = 1$

Question

Solve the problem.
-Suppose that two microphones 2,800 m apart at points A = (1,400, 0) and B = (-1,400, 0) detect the sound of a rifle shot. The time difference between the sound detected at A and the sound detected at B is 2 sec. If sound travels at approximately 330 m/sec, find an equation of the hyperbola with foci
At A and B defining the points where the shooter may be located. 
A) $\frac { x ^ { 2 } } { 330 ^ { 2 } } - \frac { y ^ { 2 } } { 1,361 ^ { 2 } } = 1$
B) $\frac { x ^ { 2 } } { 1,400 ^ { 2 } } - \frac { y ^ { 2 } } { 330 ^ { 2 } } = 1$
C) $\frac { x ^ { 2 } } { 2,800 ^ { 2 } } - \frac { y ^ { 2 } } { 2,721 ^ { 2 } } = 1$
D) $\frac { y ^ { 2 } } { 1,361 ^ { 2 } } - \frac { x ^ { 2 } } { 330 ^ { 2 } } = 1$

Quizplus · Accepted Answer

The difference in distance traveled by the sound to the two microphones is the key to finding the correct equation. Given the speed of sound is 330 m/s and the time difference is 2 seconds, the difference in distance is $330 \times 2 = 660$ meters. This distance is the major axis (2a) of the hyperbola, so $a = 330$. The foci are 2,800 meters apart, which is the distance 2c, so $c = 1,400$. Using the relationship $c^2 = a^2 + b^2$, we find $b^2 = c^2 - a^2 = 1,400^2 - 330^2 = 1,961,000 - 108,900 = 1,852,100$, so $b = 1,361$. The correct equation, with the x-axis along the line connecting the microphones and the origin at the midpoint between them, is $\frac{x^2}{330^2} - \frac{y^2}{1,361^2} = 1$.

Solve the Problem x23302−y21,3612=1\frac { x ^ { 2 } } { 330 ^ { 2 } } - \frac { y ^ { 2 } } { 1,361 ^ { 2 } } = 13302x2​−1,3612y2​=1

Solve the Problem $\frac { x ^ { 2 } } { 330 ^ { 2 } } - \frac { y ^ { 2 } } { 1,361 ^ { 2 } } = 1$